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Geometry and Topology in the Fractional Quantum Hall Effect

The FQHE is exhibited by electrons moving on a 2D surface
through which a magnetic flux passes, giving rise to

flat bands with extensive degeneracy (Landau
levels). The degeneracy

of a partially-filled Landau level is lifted by Coulomb
repulsion between the electrons, which at certain rational fillings, leads
to gapped incompressible
topologically-ordered fluid states exhibiting the FQHE. Successful model wavefunctions for FQHE states, such as the Laughlin and
Moore-Read states, are surprisingly related to Euclidean conformal field
theory, even though they are gapped incompressiible quantum fluids with a
fundamental unit of area set by the area per magnetic flux quantum h/e.

The model wavefunctions are parametrized by a
continuously-variable Euclidean metric,
just like the Euclidean conformal group of the cft to which they are related.

This metric is fixed locally both by the form of the
projected Coulomb interaction within the partially-filled Landau level, and by
local gradients of the tangential electric field on the 2D surface, promoting
it from a static flat metric fixed globally by the cft, to a dynamic local
physical degree of freedom of the FQHE fluid with area-preserving zero-point fluctuations
that leave an imprint in the ground-state structure function.

The curious connection to cft appears to be that the Virasoro algebra plays a fundament role in
both cft and FQHE, for apparently-unrelated reasons. In the FQHE it
derives from a chiral
"gravitional"

(geometric) topologically-protected anomaly at the edge of the fluid that is also
revealed in the entanglement spectrum of a cut through the bulk fluid.